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Closed-Loop Stackelberg Strategy for Linear-Quadratic Leader-Follower Game

Hongdan Li, Juanjuan Xu, Hunashui Zhang

Abstract

This paper is concerned with the closed-loop Stackelberg strategy for linear-quadratic leader-follower game. Completely different from the open-loop and feedback Stackelberg strategy, the solvability of the closed-loop solution even the linear case remains challenging. The main contribution of the paper is to derive the explicitly linear closed-loop Stackelberg strategy with one-step memory in terms of Riccati equations. The key technique is to apply the constrained maximum principle to the leader-follower game and explicitly solve the corresponding forward and backward difference equations. Numerical examples verify the effectiveness of the results, which achieves better performance than feedback strategy.

Closed-Loop Stackelberg Strategy for Linear-Quadratic Leader-Follower Game

Abstract

This paper is concerned with the closed-loop Stackelberg strategy for linear-quadratic leader-follower game. Completely different from the open-loop and feedback Stackelberg strategy, the solvability of the closed-loop solution even the linear case remains challenging. The main contribution of the paper is to derive the explicitly linear closed-loop Stackelberg strategy with one-step memory in terms of Riccati equations. The key technique is to apply the constrained maximum principle to the leader-follower game and explicitly solve the corresponding forward and backward difference equations. Numerical examples verify the effectiveness of the results, which achieves better performance than feedback strategy.
Paper Structure (8 sections, 6 theorems, 70 equations)

This paper contains 8 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Main Result
  4. Results of the follower
  5. Results of the leader
  6. The determination of $\bar{K}_{k}$ and $\bar{G}_{k}$
  7. Simulation
  8. Conclusion

Key Result

Lemma 1

Suppose the above constrained LQR problem admits a solution, then, the controller satisfies where $\lambda_k$ is as and $\beta_{k}$ is arbitrary to be determined.

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Lemma 2
  • Theorem 1
  • proof
  • Remark 4
  • Lemma 3
  • ...and 5 more